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تسجيل مستخدم جديدSparse halves in $K_4$-free graphs
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A conjecture of Chung and Graham states that every $K_4$-free graph on $n$ vertices contains a vertex set of size $lfloor n/2 rfloor$ that spans at most $n^2/18$ edges. We make the first step toward this conjecture by showing that it holds for all regular graphs.
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Kostochka and Yancey resolved a famous conjecture of Ore on the asymptotic density of $k$-critical graphs by proving that every $k$-critical graph $G$ satisfies $|E(G)| geq (frac{k}{2} - frac{1}{k-1})|V(G)| - frac{k(k-3)}{2(k-1)}$. The class of graph
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Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every pro
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Recently, Kostochka and Yancey proved that a conjecture of Ore is asymptotically true by showing that every $k$-critical graph satisfies $|E(G)|geqleftlceilleft(frac{k}{2}-frac{1}{k-1}right)|V(G)|-frac{k(k-3)}{2(k-1)}rightrceil.$ They also characteri
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